ON COMPRESSIBLE MATERIALS CAPABLE OF SUSTAINING AXISYMMETRICAL SHEAR DEFORMATIONS .2. ROTATIONAL SHEAR OF ISOTROPIC HYPERELASTIC MATERIALS

It is well known that rotational shear deformations, though not univer sal, are controllable in specific kinds of compressible and incompress ible materials. For incompressible materials, it is only necessary to identify a specific material, such as a Mooney-Rivlin material, to det ermine the rotational shear displacement function. For compressible ma terials, however, rotational shear deformations may not be possible fo r a specified class of hyperelastic materials unless certain auxiliary conditions on the strain energy function are satisfied. Polignone and Horgan have recently derived two conditions in the form of nonlinear ordinary differential equations necessary for hyperelastic materials t o sustain rotational shear deformations. In the present paper, under t he condition that the shear response function be positive, we present a single, essentially algebraic, condition necessary and sufficient to determine whether a class of compressible, homogeneous and isotropic hyperelastic materials is capable of sustaining rotational shear defor mations by application of surface tractions alone. Several examples il lustrate the simplicity of the result in applications. We have recentl y presented similar kinds of results on axisymmetric, anti-plane shear deformations. With the aid of an auxiliary necessary condition on the strain energy function, here we present a simple necessary and suffic ient condition for which both anti-plane shear and rotational shear de formations are separately possible in the same material subclass. Thes e algebraic conditions are illustrated in several examples.